The Hidden Symmetries of Yang-Mills Theory in (3+1)-dimensions

Abstract: We show that classical, non-supersymmetric Yang-Mills theories coupled to spin-1/2 and spin-0 elementary matter fields, in (3+1)-dimensional Minkowski space-time, possess an exact integrability structure with an infinite number of conserved charges in involution. Such an integrability lives on the space of non-abelian electric and magnetic charges, and is based on flat connections in generalized loop spaces, presenting an R-matrix, and Sklyanin relation. We present two novel symmetries of Yang-Mills theories. The first one corresponds to global transformations generated by the infinity of those conserved charges under the Poisson brackets. The gauge and matter fields, as well as Wilson lines and fluxes have interesting transformation laws under such a global symmetry. The second one corresponds to symmetries of the integral Yang-Mills equations, which lead to the conserved charges. They generate an infinite dimensional group, where the elements are holonomies of connections on the loop space of functions from the circle $S^1$ to the space-time. Our approach certainly applies to the Standard Model of the Fundamental Interactions. The conserved charges are gauge invariant, and so, in the case of QCD they are color singlets, and perhaps are not confined. Therefore, the hadrons may carry such charges. Our results open up the way for the construction of non-perturbative methods for Yang-Mills theories.